Thermomagnetic properties and its effects on Fisher entropy with Schioberg plus Manning-Rosen potential (SPMRP) using Nikiforov-Uvarov functional analysis (NUFA) and supersymmetric quantum mechanics (SUSYQM) methods

Thermomagnetic properties, and its effects on Fisher information entropy with Schioberg plus Manning-Rosen potential are studied using NUFA and SUSYQM methods in the presence of the Greene-Aldrich approximation scheme to the centrifugal term. The wave function obtained was used to study Fisher information both in position and momentum spaces for different quantum states by the gamma function and digamma polynomials. The energy equation obtained in a closed form was used to deduce numerical energy spectra, partition function, and other thermomagnetic properties. The results show that with an application of AB and magnetic fields, the numerical energy eigenvalues for different magnetic quantum spins decrease as the quantum state increases and completely removes the degeneracy of the energy spectra. Also, the numerical computation of Fisher information satisfies Fisher information inequality products, indicating that the particles are more localized in the presence of external fields than in their absence, and the trend shows complete localization of quantum mechanical particles in all quantum states. Our potential reduces to Schioberg and Manning-Rosen potentials as special cases. Our potential reduces to Schioberg and Manning-Rosen potentials as special cases. The energy equations obtained from the NUFA and SUSYQM were the same, demonstrating a high level of mathematical precision.

where D is the potential depth, α is the screening parameter and δ 0 is the potential parameter that determines the size of the potential and can also serve as optimizing parameter. Recently, there has been a surge of interest in incorporating at least two potentials. The goal of combining at least two potential models is to provide more physical application and analysis to existing molecular physics studies. Also, it is well-known that the potential energy functions with more parameters have a tendency to fit experimental data better than those with fewer parameters [27][28][29] . Many scholars have conducted extensive research in both relativistic and non-relativistic regimes to explore these potentials [30][31][32][33][34][35][36][37][38] .
In recent times, research indicates that the addition of external fields to potential functions on quantum systems has demonstrated its potency in controlling certain behaviors of systems and molecules 39 . The Aharonov-Bohm (AB) effect, discovered in 1961 40 , occurs when a moving charge is transformed by scalar and vector potentials that appear in the Schrödinger equation (SE) even in the absence of external EM fields 41 . Since then, many studies have analyzed a bound state of a charged particle moving in a potential vector and scalar potential. A realistic description of the external EM field effects on quantum systems is provided by the Stark 42 and Zeeman 43 effects. In the Stark effects, an external electric field is applied to the electrically neutral hydrogen atom, causing it to experience a zero net force, resulting in a shift in the energy levels. On the other hand, Zeeman effects occur when an atom is exposed to a uniform magnetic field. These interactions have similar effects in that they cause the energy levels to split and shift 44 . External fields have previously been studied by a wide range of quantum mechanical phenomena in many areas, including physics, chemistry, biology, material science, engineering, mathematics [45][46][47][48][49][50][51] and others.
Considering the vast applicability of the Manning-Rosen and Schioberg potentials, it is necessary to investigate the bound state solutions of the two-dimensional (2D) SE with the combined potential under the influence of external magnetic and Aharonov-Bohm (AB) fields and their effects on the Shannon entropy and Fisher information for some selected diatomic molecules. The bound state solutions will be obtain using the Nikiforov-Uvarov-Functional Analysis (NUFA) and supersymmetric quantum mechanics (SUYSQM) methods.
This paper is organized as follows: first, we provide detailed solutions to the 2D SE with Manning-Rosen plus Schioberg potential (SPMR) in the presence of magnetic and Aharonov-Bohm (AB) flux fields using the NUFA method. Second, we used the SUYSQM method to obtain the analytical solution of the SE with the combined potential in the presence of magnetic and Aharonov-Bohm (AB) flux fields. Also, the normalized wavefunction obtained is applied to investigate the Shannon entropy and Fisher information in the presence and absence of external magnetic and Aharonov-Bohm (AB) flux fields. Finally, the concluding remarks. The SPMR is of the form.

Nikiforov-Uvarov-Functional Analysis (NUFA) method
The Nikiforov-Uvarov Functional Analysis (NUFA) method recently developed by Ikot et al. 52 has been very helpful in providing solutions for exponential type potentials both in relativistic and nonrelativistic wave equations When using this method to solve either the Schrödinger or Klein-Gordon equation, the energy eigen equation is directly presented in a factorized, closed and compact form. This gives the method an edge over other methods. Meanwhile, the NUFA theory involves solving second order Schrödinger-like differential equation through the analytical combination of Nikiforov-Uvarov (NU) method and functional analysis approach [53][54][55] . NU is applied to solve a second-order differential equation of the form where σ (s) and σ (s) are polynomials at most degree two and τ (s) is a first-degree polynomial. Tezean and Sever 56 latter introduced the parametric form of NU method in the form whereα i and ξ i (i = 1, 2, 3) are all parameters. The differential Eq. (3) has two singularities which is at s → 0 and s → 1 α 3 thus, the wave function can be expressed in the form.
Substituting Eq. (6) into Eq. (5) and simplifying culminate to the following equation, Equation (7) can be reduced to a Guassian-hypergeometric equation if and only if the following functions vanished Applying the condition of Eq. (8) and Eq. (9) into Eq. (7) results into Eq. (10) The solutions of Eqs. (8) and (9) are given as  Setting either a or b equal to a negative integer -n, the hypergeometric function f(s) turns to a polynomial of degree n. Hence, the hypergeometric function f(s) approaches finite in the following quantum condition, i.e.,a = −n where n = 0, 1, 2, 3 . . . n max or b = −n.
Using the above quantum condition, By simplifying Eq. (18), the energy eigen equation using NUFA method is given as By substituting Eqs. (9) and (10) into Eq. (6), the corresponding wave equation for the NUFA method as Thermomagnetic energy spectra of 2-dimensional Schrödinger equation under the influence Aharanov-Bohm (AB) flux and external magnetic field using NUFA. The thermomagnetic energy spectra of 2-Dimensional Schrödinger equation under the influenced of AB and Magnetic field with SPMR potential can be obtained from charged particle Hamiltonian operator of the form E nm is the thermomagnetic energy spectra, e and µ represent the charge of the particle and the reduced mass respectively. c is the speed of light. Meanwhile, The vector potential − → A = A r , A φ , A z can be written as the superposition of two terms such that The vector potential − → A can then be expressed as The Laplacian operator and the wave function in cylindrical coordinate is given as where ξ = � AB φ 0 is an absolute value containing the flux quantum φ 0 = hc e . The cyclotron frequency is represented by ω c = e � B µc . Equation (24) is not exactly solvable due to the presence of centrifugal barrier 1 r 2 . In order to provide an analytical approximate solution to Eq. (24), we substitute the modified Greene-Aldrich approximation of the form 1 (1−e −αr ) 2 into Eq. (24) to deal with the centrifugal barrier. Also, using the coordinate transformation s = e −αr together with the approximation term, Eq. (24) reduced to the hyper-geometric equation of the form where Comparing Eq. (25) with NUFA differential equation in Eq. (5), the following polynomial equations can be obtained.

Thermomagnetic energy spectra of 2-dimensional Schrodinger equation under the influence Aharanov-Bohm (AB) flux and external magnetic field using super symmetric quantum mechanics approach
The supersymmetric approach deals with partner Hamiltonian of the form where p is the momentum and V (x) is the effective potential. The effective potential can be expressed in terms of super potential as The ground state energy is obtained as www.nature.com/scientificreports/ where N is the normalization constant which for a very simple case can be determined using the expression However, the super potential satisfies the shape invariance condition where a 1 is a new set of parameter determines from the old set a 0 through the mapping f : a 0 → a 1 = f (a 0 ). The total supersymmetric energy is defined as While higher order state solutions are obtained through the expression where A + (a k ) is a raising ladder operator expressed as Also, the Schrodinger equation under super symmetric quantum mechanics approach is arranged in the form With the help of approximation to centrifugal term, Eq. (24) can be re-arranged as follows Equation (53) can then be compared to Eq. (52) such that The proposed super potential that is suitable for the effective potential is given as The supersymmetric partner potential can be obtained as follows : Equation (56) obeys shape invariance condition The excited state energy is calculated using shape invariance condition If g = g 0 , g 1 = g 0 + 1 , g n = g 0 + αn . Then using Eq. (67), then, the shape invariance condition equation become (59) Recall that g = g 0 , g n = g 0 + αn = g + αn . Using Eq. (74), the higher order supersymmetric energy can be evaluated as Meanwhile, the total energy is the ground state energy plus higher order supesymmetric energy Substituting Eqs. (61) and (75)   www.nature.com/scientificreports/ Using the supersymmetric mapping g n :→ g + αn with the total energy expressed as Ẽ nm = 2µE nm

Thermomagnetic properties
The thermodynamic properties of quantum systems can be obtained from the exact partition function given by where, is an upper bound of the vibrational quantum number obtained from the numerical solution of dE n dn = 0 , given as = −δ + √ Q 3 , β = 1 kT where k and T are Boltzmann constant and absolute temperature respectively. In the classical limit, the summation in Eq. (82) can be replaced with an integral: Using Eq. (83), the partition function can be expressed as

Results and discussion
Figures 1a-d are the plots of variation of thermomagnetic energy spectra against the screening parameter in the absence of AB and magnetic field, the presence of the only magnetic field, the presence of only AB field and the presence of both magnetic and AB fields, respectively. In Fig. 1a-d, the bound state energy spectral diagrams all increases monotonically with increasing values of the screening parameter ( α ) in such a unique and quantized manner. Figure 2a, b are the variation of wave function plot against the radial distance in the absence of both AB and magnetic field and the variation of probability density plot against the radial distance in absence of both AB and magnetic field, respectively. In Fig. 2a, the wave function showcases intertwining multiple sinusoidal curves representing the different quantum states. In Fig. 2b, the probability density plots in the absence of AB and magnetic field show a normal distribution curve with multiple peaks, each depicting a different quantum state. It is interesting to note that in Fig. 2a, the ground state has the lowest peak, while the highest state ( n = 3 ) has the highest peak. Figure 2b agreed excellently with the theoretical and experimental descriptions of probability density. It is expected that in an ideal condition, the peak of the probability density plot should increase as the quantum state increases. This is only possible because in Fig. 2a and b, the wave function and probability density plots are carried out in the absence of AB and magnetic field respectively. Figure 3a, b are the variations of the wave function and the probability density plots against the radial distance in the presence of magnetic field. Figure 3a shows a periodic and sinusoidal wave function similar to Fig. 2a. However, in Fig. 3b, there is distortion in the probability distribution curves because of the presence of magnetic field. The presence of the magnetic field does not allow uniform distribution of probability density plots in increasing order of the quantum state whose highest peak supposed to occur at ( n = 3 ). However, in Fig. 3b, n = 2 has the highest peak, followed by n = 0 before n = 3 . The disorderliness, ambiguity and distortions in the peaks clearly show the effect of magnetic field. Figure 4a, b are the variation of the wave function and probability density plots against the radial distance in the presence of AB field, respectively. Figure 4a and b has similar explanation to Fig. 3a and b when the distortions to the probability density plot are affected by the presence of Aharonov-Bohm flux field. Figure 5a, b show how the wave function and probability density varied with radial distance in the presence of both magnetic and AB fields . Under the influence of AB and magnetic fields, the wave function in Fig. 5a is sinusoidal and periodic. .However, in Fig. 5b, something fascinating occurs. The peaks of probability density plot for quantum state ( n = 1 ) are almost the same as n = 2 , i.e., the combined effect of AB and magnetic effect establish quantum state equivalence. Figure 6a-d are the plot of partition function against magnetic flux (ω c ) for different values of inverse temperature parameter (β) , plot of partition function against AB flux (ξ ) for different values of inverse temperature parameter ( β ), plot of partition function against inverse temperature parameter ( β ) for fixed value of ω c and ξ but for different values of maximum vibrational quantum number ( ) and plot of partition function against the maximum vibrational quantum number ( )for fixed value of ω c and ξ but for different values of inverse temperature parameter ( β ), respectively. In Fig. 6a, the partition function starts from the negative y-axis an increase exponentially with increasing value of the magnetic field. The same explanation occurs in Fig. 6d where the partition function increases exponentially with an increase in maximum vibrational quantum number.
(105) � 0 = 86 + 94β − 64β 2 + 42(47 + 56β)η + 4(43 + 30β)η 2 + 16(5 + β)η 3 + 16η 4  Fig. 6b, the partition function rises monotonically with unique spacing before reaching a peak value with local maximum turning point at ξ = 40 . In Fig. 6c, the partition function increases monotonically with an increase in inverse temperature parameter. Figure 7a-d are the plot of vibrational mean energy against magnetic flux (ω c ) for different values of inverse temperature parameter (β) , plot of vibrational mean energy against AB flux (ξ ) for different values of inverse temperature parameter ( β ), plot of vibrational mean energy against inverse temperature parameter ( β ) for fixed value of ω c and ξ but for different values of maximum vibrational quantum number ( ) and plot of vibrational mean energy against the maximum vibrational quantum number ( ) for fixed value of ω c and ξ but for different values inverse temperature parameter ( β ) respectively.
In Fig. 7a, the vibrational mean energy showcase a parabolic curve which increases with an increase in magnetic field. In Fig. 7b, the vibrational mean energy increases monotonically before converging at ξ = 6 with increase in AB flux. Also, the vibrational mean energy increases uniquely from the origin with quantized spacing, in an increasing value of inverse temperature parameter as shown in Fig. 7c. Correspondingly, the vibrational    www.nature.com/scientificreports/ heat capacity against the maximum vibrational quantum number ( ) for fixed value of ω c and ξ but for different values inverse temperature parameter ( β ) respectively. In Fig. 8a, the vibrational heat capacity increases monotonically with increase in magnetic field. In Fig. 8b, the vibrational heat capacity shows symmetrical curves with common converged maximum point at ξ = 45 . This maximum point divides the curves into equal half both in a decreasing and increasing value of ξ . The physical interpretation of Fig. 8b is that heat capacity from the concept of molecular vibration relates to the kinetic energy of the molecules of the system. So, the Fig. 8b completely shows that with the influence of Aharanov-Bohm flux field, the kinetic energy of the molecules of the system remains constant during molecular vibrations. This explains why there is a symmetrical curves both at the left and right hand side of the thermomagnetic plot. In Fig. 8c, the vibrational heat capacity is a parabolic curve that concaves upward with minimum turning point at β = 0.0005 K −1 before rising to various local maximum turning points in increasing value of β , before converging at β = 0.004 K −1 . In Fig. 8d, the specific heat capacity increases asymmetrically to various unique maximum point before converging at = 1000 with increasing value of maximum vibrational quantum number.  Fig. 9a and d, the vibrational entropy increases exponentially with an increase in magnetic field and maximum vibrational quantum number respectively. In Fig. 9b, the vibrational entropy rises to the peak with maximum turning point at ξ = 35 before slopping in a divergence manner with distinct spacing between the spectral curves. In Fig. 9c, the vibrational entropy increases exponentially with an increase in inverse temperature parameter.   for different values of inverse temperature parameter (β) , plot of magnetization against AB flux (ξ ) for different values of inverse temperature parameter ( β ), plot of magnetization against inverse temperature parameter ( β ) for fixed value of ω c and ξ but for different values of maximum vibrational quantum number ( ) and plot of magnetization against the maximum vibrational quantum number ( ) for fixed value of ω c and ξ but for different values inverse temperature parameter ( β ) respectively. In Fig. 11a, c and d, the magnetization increases exponentially with an increase in ω c , β and respectively. However, in Fig. 11b the influence of AB field produces normal distribution curves with distinct peaks corresponding to the values of inverse temperature parameter ( β).  www.nature.com/scientificreports/ temperature parameter ( β ), plot of magnetic susceptibility against inverse temperature parameter ( β ) for fixed value of ω c and ξ but for different values of maximum vibrational quantum number ( ) and plot of magnetic susceptibility against the maximum vibrational quantum number ( ) for fixed value of ω c and ξ but for different values inverse temperature parameter ( β ) respectively. In Fig. 12a, the magnetic susceptibility increases monotonically from zero into diverging curves. In Fig. 12b, the magnetic susceptibility produces sinusoidal curves with discontinuity at ξ = 50 . In Fig. 12c, the magnetic susceptibility rises to attain various local maximum point at precisely β = 0.125 K −1 . Also, in Fig. 12d, the magnetic susceptibility increases exponentially with an increase in maximum vibrational quantum number.  Fig. 13c plot of persistent current against inverse temperature parameter ( β ) for fixed value of ω c and ξ but for different values of maximum vibrational quantum number ( ) and plot of persistent current against the maximum vibrational quantum number ( ) for fixed value of ω c and ξ but for different values inverse temperature parameter ( β ) respectively. In Fig. 13a and d, the persistent current increases asymptotically from the origin with increasing value of magnetic field and maximum upper bound vibrational quantum number. In Fig. 13b, the persistent current rises from the origin to exhibits various maximum points before concaving upward with unique minimum points with maximum at ξ = 45 . In Fig. 13c, the persistent current increases from the vertical axis in a quantized form before diverging into various spectral curves with increasing value of β.  www.nature.com/scientificreports/ and momentum space Fisher entropy against the screening parameter for n = 0 respectively. In Fig. 14a, the position space Fisher entropy increases linearly with an increase in the screening parameter, while the momentum space and its product increases exponentially with an increase in the screening parameter ( α ) as shown in Fig. 14b and c respectively.   Figure 16a-c has the same explanation as Fig. 14a-c. Figure 17a-c are the plot of position space Fisher entropy against the screening parameter for n = 3, the plot of momentum space Fisher entropy against the screening parameter for n = 3 and the plot of product of position and momentum space Fisher entropy against the screening parameter for n = 3 respectively. In Fig. 17a, the position space entropy increases exponentially with an increase in the value of α . However, in Fig. 17b and c, there is abnormally which makes the plot to decrease with decreasing value of α with respect to momentum space and its products respectively. Table 1 is the numerical bound state solution for the proposed potential under the influence of AB and Magnetic field for fixed magnetic quantum number but with varying principal quantum number. In Table 1, it can be observed that when both fields are deactivated, i.e., AB and magnetic fields are zero, the energy spectra degenerate; thus, as the number of quantum states increases, the energy spectra decrease. When only the AB field was applied to the quantum system, it resulted in quasi-degeneracy, and the energy spectra decreased with www.nature.com/scientificreports/ increasing quantum states. When only the magnetic field is activated, the system produces a similar effect, but this time degeneracy is gradually eliminated. When both fields are activated, the combined effects completely eliminates degeneracy from the quantum system's energy spectra. All computation were carried out using the following constant physical parameters: c 1 = c 2 = 1, σ 0 = 0.5, = µ = 1, α = 0.2, c = 1. Tables 2, 3, 4 and 5 are the numerical computation for position, momentum, and products Fisher entropy under the influence of AB and magnetic field for n = 0 to n = 3 , respectively. In these Tables, it is clear that our results obey Heisenberg uncertainty principles in which there is uncertainty in the simultaneous measurement of the position and momentum of quantum mechanical particles. The numerical results also show that as the values This trend holds for all quantum states in the absence of both magnetic and AB fields, only magnetic fields, only AB fields, and the combined influence of both magnetic and AB fields. Correspondingly, our numerical results in all quantum states satisfy the 2D local Fisher uncertainty product inequality expressed as ( I(ρ)I(γ ) ≥ 16 as shown in Tables 2, 3, 4 and 5 for all quantum states. All our results clearly show that as the quantum state increases, the values of position increases, while that of momentum and product values decrease. The Fisher product values in all quantum states clearly show the localization of the quantum mechanical particles both in the absence and presence of magnetic and AB fields. Finally, the numerical results from their product indicate that the particle is more localized when the combined effect of AB and magnetic fields on the entropy than the absence of both fields, as shown by ( ( I(ρ)I(γ ) ≥ 16).

Conclusion
In this work, we study analytical solutions, thermomagnetic properties, and its effect on Fisher information entropy with Schioberg plus Manning-Rosen potential using the Nikiforov-Uvarov functional analysis and Supersymmetric quantum mechanics methods. We obtained the energy equation in a closed and compact form both in NUFA and SUSYQM and applied the solution to study partition function and other thermomagnetic properties.
The trend of thermomagnetic plots is in excellent agreement with the work of existing literature. Using the normalized wave function, we obtained the wave function and probability density plots and applied them to study Fisher information entropy in position and momentum spaces. The numerical results show that the combined impact of the magnetic and AB flux fields completely removes the degeneracy on the energy spectra and that increasing the screening parameter increases the position of Fisher entropy while decreasing its momentum, satisfying the 2D local Fisher uncertainty product condition. It also causes both localization and delocalization of quantum particles. Meanwhile, as the quantum state increases under the combined influence of magnetic and www.nature.com/scientificreports/ AB fields, the results of Fisher entropies and the product increase. Finally, the proposed potential reduces to Schioberg and Manning-Rosen potential as special cases. The wave function and probability density plots were obtained using Maple 10,0 software, while the position and momentum Fisher entropies were obtained using a well-designed Mathematica program.

Data availability
The data available in this manuscript are obtained using maple and Mathematica programme from the resulting energy eigen equation.        Table 3. Numerical values for position, momentum and products Fisher entropy under the influence of AB and Magnetic field for n = 1.